**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Course# MATH-488: Algebraic K-theory

Summary

Algebraic K-theory, which to any ring R associates a sequence of groups, can be viewed as a theory of linear algebra over an arbitrary ring. We will study in detail the first two of these groups and applications of algebraic K-theory to number theory, algebraic topology, and representation theory.

Moodle Page

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related MOOCs (9)

Related concepts (77)

Algebra (part 1)

Un MOOC francophone d'algèbre linéaire accessible à tous, enseigné de manière rigoureuse et ne nécessitant aucun prérequis.

Algebra (part 1)

Un MOOC francophone d'algèbre linéaire accessible à tous, enseigné de manière rigoureuse et ne nécessitant aucun prérequis.

Algebra (part 2)

Un MOOC francophone d'algèbre linéaire accessible à tous, enseigné de manière rigoureuse et ne nécessitant aucun prérequis.

Algebraic K-theory

Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called K-groups. These are groups in the sense of abstract algebra. They contain detailed information about the original object but are notoriously difficult to compute; for example, an important outstanding problem is to compute the K-groups of the integers.

Ring (mathematics)

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

Algebraic number theory

Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations.

K-theory

In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It is also a fundamental tool in the field of operator algebras. It can be seen as the study of certain kinds of invariants of large matrices.

Zariski topology

In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in real or complex analysis; in particular, it is not Hausdorff. This topology was introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring (called the spectrum of the ring) a topological space.

Related courses (11)

MATH-311: Algebra IV - rings and modules

Ring and module theory with a major emphasis on commutative algebra and a minor emphasis on homological algebra.

MATH-334: Representation theory

Study the basics of representation theory of groups and associative algebras.

MATH-506: Topology IV.b - cohomology rings

Singular cohomology is defined by dualizing the singular chain complex for spaces. We will study its basic properties, see how it acquires a multiplicative structure and becomes a graded commutative a

MATH-110(a): Advanced linear algebra I

L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et de démontrer rigoureusement les résultats principaux de ce sujet.

MATH-535: Algebraic geometry III - selected topics

This course is aimed to give students an introduction to the theory of algebraic curves, with an emphasis on the interplay between the arithmetic and the geometry of global fields. One of the principl